# A man repays a loan of $3250 by paying$20 in the first month and then increases the payment by $15 every month. How long will it take him to clear the loan? ##### 1 Answer Nov 29, 2016 Define ${p}_{k}$as the payment in the month $k + 1$. We have: p_0=20$
p_k=20$+k* 15$ for $k = 1 , 2 , \ldots$

At the end of the n-th month the total payment is:

${P}_{n - 1} = 20 + {\sum}_{1}^{n - 1} {p}_{k} = 20 + {\sum}_{1}^{n - 1} \left(20 + 15 k\right) = 20 n + 15 {\sum}_{1}^{n - 1} k$

Using Gauss' formula for the sum of the first $\left(n - 1\right)$ integers:

${P}_{n - 1} = 20 n + 15 \frac{n \left(n - 1\right)}{2}$

express this as an equation in $n$ and pose P_n=3250$ $3250 = 20 n + 15 \frac{n \left(n - 1\right)}{2}$Solve for $n$: $15 {n}^{2} - 15 n + 40 n - 6500 = 0$$15 {n}^{2} + 25 n - 6500 = 0$$n = \frac{- 25 \pm \sqrt{625 + 4 \cdot 15 \cdot 6500}}{30} = \frac{- 25 \pm 625}{30}$Obviously we discard the negative solution and $n = \frac{600}{30} = 20\$

So the debt is repaid at the end of the 20-th month.